What are Even Numbers and Its Different Types

An integer is considered an 'even number' if it is a multiple of 2 or, when divided by 2, it leaves no remainder. Let us understand even numbers with two examples. While the JEE Main and IIT JAM exams do not directly ask questions from this topic, it is advisable to understand the concept to build a solid mathematical foundation. 

A box has 12 bananas. Are these bananas even in number? Let us understand how we can identify it.

1. The first condition is that the integer should be a multiple of 2. If we check the table of 2, you will notice that 6 times 2 is 12. So, the first condition has been fulfilled.

2. Now, we need to check if the number of bananas is completely divisible by 2. It means that no remainder is left. For this, we will start grouping bananas in pairs of 2. As we pair the bananas, we notice that six pairs are there and no banana has been left alone. In simple words, the remainder is zero since zero bananas have been left alone. This fulfils the second condition.

3. Let us check for the third condition, which says that when an even number is added to another even number, it is still an even number. Let us add two more bananas to the 12 bananas. Is the final number of bananas even? Fourteen bananas are even since 14 is a multiple of 2 (7 x 2 = 14). On dividing 14 by 2, the remainder is 0. If I start grouping 14 bananas in pairs of 2, no banana will be left alone, i.e., the remainder is 0. This fulfils the third condition.

Let us discuss different types of even numbers:

• Consecutive Even Numbers: These are the even numbers that are in either ascending or descending order. For instance, 2,4,6,8,10,12,14 or 20,22,24,26,28 are consecutive even numbers. These are consecutive since there is no other even number missing in the series, and these are in ascending order.
• Even natural numbers: These are the types of even numbers that belong to natural numbers. This means 0 will not be an even natural number. These even numbers can be in any order.

Let us now understand the properties of even numbers:

Property of Addition

Let us take a look at the different properties applicable to these numbers:

1. a(even number) + b(even number) = even number

On adding two even numbers, the resulting sum is also an even number. Suppose a and b are two even numbers. On adding a and b,

For eg: a = 2 and b = 8

2 + 8 = 10 

10 is also an even number since it is completely divisible by 2. 

2. a(odd number) + b(even number) = odd number

On adding an even number with an odd number or vice-versa, the resulting sum will always be an odd number.

Say a = 5 and b = 10

5 + 10 = 15

15 is an odd number since it is not completely divisible by two and leaves one as a remainder.

Properties of Subtraction

Let us take a look at the property of subtraction for even numbers:

1. x(even number) - y(even number) =  even number

On subtracting two even numbers, the resulting difference between the two numbers will also be an even number.

Say, x = 14 and y = 10

14 - 10 = 4

4 is an even number since it is a multiple of 2 (2 x 2 =4)

2. x(odd number) - y(even number) = (odd number)

or

x (even number) - y (odd number) = odd number

On subtracting an even number from an odd number, the resulting difference will always be an odd number. 

x = 15 and y = 8

15 - 8 = 7

7 is an odd number since it is neither the multiple of 2 nor it is completely divisible by 2.

Now, let us say that x = 16 and y = 5

16 - 5 = 11 

11 is also an odd number since it is not completely divisible by 2.

Properties of Multiplication

Let us take a look at the properties of multiplication

1. p(even number) * q(even number) = even number

On multiplying two even numbers, the resulting product will also be an even number.

p = 6 and q = 4

6 x 4 = 24

24 is completely divisible by 2, which makes it an even number.

2. p(odd number) * q(even number) = even number

On multiplying an odd number by an even number, the resulting product will always be an even number.

p = 7 and q =12

7 x 12 = 84

84, again is an even number, since it is completely divisible by 2 and leaves no remainder.

Only one number is categorised as an even prime to date. The number is 2, which is both even and prime. Let us show you how number 2 is both.

1. Proving 2 is an even number.

• When you divide 2 by 2, it leaves no remainder. 
• As you look at 2, you notice that it is already a multiple of 2 (1 x 2 = 2).
•  Adding any odd number to 2 will result in an odd number. This is the additive property of even numbers. For example, 2 + 5 = 7, which is an odd number.

2. Proving 2 is a prime number.

• 2 is divisible by 1.
• Also, number 2 is divisible by itself.
• However, number 2 is not divisible by any other number. This makes number 2 a prime number.